An upper bound on the time required to implement unitary operations

Abstract

We derive an upper bound for the time needed to implement a generic unitary transformation in a d dimensional quantum system using d control fields. We show that given the ability to control the diagonal elements of the Hamiltonian, which allows for implementing any unitary transformation under the premise of controllability, the time T needed is upper bounded by T ≤ πd 2 (d−1) 2g min where gmin is the smallest coupling constant present in the system. We study the tightness of the bound by numerically investigating randomly generated systems, with specific focus on a system consisting of d energy levels that interact in a tight-binding like manner.

From Pulses to Circuits and Back Again: A Quantum Optimal Control Perspective on Variational Quantum Algorithms

“…Exact results are only known for certain systems consisting of 1-3 qubits [125][126][127][128][129]. However, upper and lower bounds on T can be found [130][131][132][133]. In addition, for n qubit networks, the upper-bounds in [132,133] allow for characterizing the unitary transformations that can be created efficiently with 2n local fields, i.e., where T = O(poly(n)).…”

“…However, upper and lower bounds on T can be found [130][131][132][133]. In addition, for n qubit networks, the upper-bounds in [132,133] allow for characterizing the unitary transformations that can be created efficiently with 2n local fields, i.e., where T = O(poly(n)). While some progress has recently been made to characterize efficiently-controllable qubit graphs [134], in general, it remains an open challenge to systematically determine the set of unitary transformations that are reachable in polynomial time with fewer controls.…”

Algorithmic Primitives for Quantum-Assisted Quantum Control

“…Our techniques add to the literature on variational quantum algorithms [46][47][48] and generalize the optimization program to complex overlaps which bring control theory objectives and variational quantum algorithms closer [49]. Furthermore, our algorithmic primitives can also be used to optimize non-gradient objectives and can be used for the important tasks such as the design of modular quantum computers [50][51][52] and controlling reactions in quantum chemistry [53].…”